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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Almost every quasinilpotent Hilbert space operator is a universal quasinilpotent
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by Domingo A. Herrero PDF
Proc. Amer. Math. Soc. 71 (1978), 212-216 Request permission

Abstract:

Let Q be a quasinilpotent operator acting on a complex separable infinite dimensional Hilbert space; then either ${Q^k}$ is compact for some positive integer k, or the closure of the similarity orbit of Q contains every quasinilpotent operator. Analogous results are shown to be true for the Calkin algebra and for nonseparable Hilbert spaces. For the nonseparable case, the analogous result is true for the closed bilateral ideal $\mathcal {J}$, strictly larger than the ideal of compact operators, if and only if $\mathcal {J}$ is not the ideal associated with an ${\aleph _0}$-regular limit cardinal. For the ideal of compact operators, the problem remains open.
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Additional Information
  • © Copyright 1978 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 71 (1978), 212-216
  • MSC: Primary 47A65; Secondary 47B05
  • DOI: https://doi.org/10.1090/S0002-9939-1978-0500492-1
  • MathSciNet review: 500492